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Polya’s Problem Solving Techniques

Polya s Problem Solving TechniquesIn 1945 George Polya published the bookHow To Solve Itwhich quickly becamehis most prized publication. It sold over one million copies and has been translatedinto 17 languages. In this book he identifies four basic principles of Problem s First Principle: Understand the problemThis seems so obvious that it is often not even mentioned, yet studens are oftenstymied in their efforts to solve problems simply because they don t understand itfully, or even in part. Polya taught teachers to ask students questions such as: Do you understand all the words used in stating the Problem ? What are you asked to find or show? Can you restate the Problem in your own words? Can you think of a picture or diagram that might help you understand theproblem? Is there enough information to enable you to find a solution?

into 17 languages. In this book he identi es four basic principles of problem solving. Polya’s First Principle: Understand the problem This seems so obvious that it is often not even mentioned, yet studens are often stymied in their e orts to solve problems simply because they don’t understand it fully, or even in part.

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Transcription of Polya’s Problem Solving Techniques

1 Polya s Problem Solving TechniquesIn 1945 George Polya published the bookHow To Solve Itwhich quickly becamehis most prized publication. It sold over one million copies and has been translatedinto 17 languages. In this book he identifies four basic principles of Problem s First Principle: Understand the problemThis seems so obvious that it is often not even mentioned, yet studens are oftenstymied in their efforts to solve problems simply because they don t understand itfully, or even in part. Polya taught teachers to ask students questions such as: Do you understand all the words used in stating the Problem ? What are you asked to find or show? Can you restate the Problem in your own words? Can you think of a picture or diagram that might help you understand theproblem? Is there enough information to enable you to find a solution?

2 Polya s Second Principle: Devise a planPolya mentions that there are many reasonable ways to solve problems. The skillat choosing an appropriate strategy is best learned by Solving many problems. Youwill find choosing a strategy increasingly easy. A partial list of strategies is included: Guess and check Look for a pattern Make an orderly list Draw a picture Eliminate possibilities Solve a simpler Problem Use symmetry Use a model Consider special cases Work backwards Use direct reasoning Use a formula Solve an equation Be ingenious1 Polya s Third Principle: Carry out the planThis step is usually easier than devising the plan. In general, all you need iscare and patience, given that you have the necessary skills. Persist with the plan thatyou have chosen. If it continues not to work discard it and choose another. Don t bemisled, this is how mathematics is done, even by s Fourth Principle: Look backPolya mentions that much can be gained by taking the time to reflect and lookback at what you have done, what worked, and what didn t.

3 Doing this will enableyou to predict what strategy to use to solve future starting on the next page, here is a summary, in the master s own words, onstrategies for attacking problems in mathematics class. This is taken from the book,How To Solve It, by George Polya, 2nd ed., Princeton University Press, 1957, UNDERSTAND THE Problem have tounderstandthe Problem . What is the unknown? What are the data? What is the condition? Is it possible to satisfy the condition? Is the condition sufficient to deter-mine the unknown? Or is it insufficient? Or redundant? Or contradictory? Draw a figure. Introduce suitable notation. Separate the various parts of the condition. Can you write them down?2. DEVISING A PLAN the connection between the data and the unknown. Youmay be obliged to consider auxiliary problems if an immediate connectioncannot be found.

4 You should obtain eventually aplanof the solution. Have you seen it before? Or have you seen the same Problem in a slightlydifferent form? Do you know a related Problem ?Do you know a theorem that could beuseful? Look at the unknown!Try to think of a familiar Problem having the sameor a similar unknown. Here is a Problem related to yours and solved before. Could you use it?Could you use its result? Could you use its method? Should you introducesome auxiliary element in order to make its use possible? Could you restate the Problem ? Could you restate it still differently? Goback to definitions. If you cannot solve the proposed Problem , try to solve first some relatedproblem. Could you imagine a more accessible related Problem ? A moregeneral Problem ? A more special Problem ? An analogous Problem ? Couldyou solve a part of the Problem ? Keep only a part of the condition, dropthe other part; how far is the unknown then determined, how can it vary?

5 Could you derive something useful from the data? Could you think ofother data appropriate to determine the unknown? Could you change theunknown or data, or both if necessary, so that the new unknown and thenew data are nearer to each other? Did you use all the data? Did you use the whole condition? Have youtaken into account all essential notions involved in the Problem ?33. CARRYING OUT THE PLAN outyour plan. Carrying out your plan of the solution,check each step. Can you see clearlythat the step is correct? Can you prove that it is correct?4. LOOKING BACK solution obtained. Can youcheck the result? Can you check the argument? Can you derive the solution differently? Can you see it at a glance? Can you use the result, or the method, for some other Problem ?4


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